Novel development of Lyapunov stability of motion

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Abstract
The paper presents a refined analysis of the influence of initial data on dynamic behaviour and stability properties of non-stationary systems. As a result, new stability properties are discovered and defined as well as the corresponding stability domains. Furthermore, general necessary conditions are established for asymptotic stability of the unperturbed motion of these systems, the instantaneous asymptotic stability domain of which can be either time-invariant or time-varying and then possibly asymptotically contractive. It is shown that the classical Lyapunov stability conditions cannot be applied to the stability test as soon as the system instantaneous domain of asymptotic stability is asymptotically contractive. In order to investigate asymptotic stability of the motion in such cases novel criteria are established. Under the criteria the Eulerian derivative of a system Lyapunov function may be non-positive only and still guarantee asymptotic stability of the unperturbed motion The results are illustrated by examples.

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